In a tour-de-force of mathematical physics, the 1d Hubbard chain was exactly solved by the Bethe-ansatz methodThe Bethe Ansatz solution of the Hubbard model is incomplete in the sense that it affords analytic extraction of response functions only in certain limiting cases. Numerical methods (or bosonization) are still required in order to evaluate general response functions. [1]}. For this as well as more general 1d problems, such as the Hubbard model on various relatively narrow width ladders and cylinders, much more is known than for higher dimensional cases. Particularly powerful numerical methods can be deployed, as discussed in § REF  below. In addition, a weak-coupling renormalization group (RG) - known colloquially as “\(g\) -ology” in reference to the various couplings \(g^{\vphantom{\dagger }}_a\)  - leads to a rich set of results involving multiple “intertwined orders” in the sense that the interactions that promote charge-density-wave, spin-density-wave, and singlet and triplet superconducting correlations evolve in a complex, interrelated fashion under RG.
